A threefold deformation decomposition in shape analysis for medical imaging: Spherical, deviatoric and non affine components

Statistical Shape Analysis, that includes classic Geometric Morphometrics (GM), is often based on landmarks and frequently used to describe shape and shapes changes in biological studies. In this context changes in shape are typically analyzed separately from changes in size measured, in most cases, with centroid size (CS) [1]. Changes in shape are projected in a common linear space: the tangent space to the consensus and decomposed in affine and non affine components. The non affine component can be in turn de-composed in a series of local deformations (partial warps). This approach relies on the assumption that shapes are limitedly scattered in the shape space. In these conditions the difference between centroid size and m-volume is barely appreciable. In medical image, and in general in soft tissues, bodies can undergo very large deformations, involving also large changes in size strictly coupled with the change in shape from mechanical point of view. The cardiac example, analyzed in the present paper, shows changes in volume that can reach the 60% when comparing systole and diastole, coupled with severe longitudinal, radial and torsional strains. Because ventricle’s volume, together with its pressure, is one of the most important descriptors of the pumping heart function, it is natural considering the size change as volumes change. In fact, when dealing with such large size differences, CS and volume behave differently. In the last years the emerging disciplines of Diffeomorphometry and the related Functional Anatomy apply to replace classic GM through the use of diffeomorphisms, more suitable to describe soft tissues changes [2, 3]. On the other hand, these descriptions are very sophisticated but ignore some synthetic properties of the decomposition of deformation in some significant aspects. The goal of the present work is to show that standard GM tools (landmarks, Thin Plate Spline, and related decomposition of the deformation) can be generalized to better describe the very large deformations of biological tissues, without losing a synthetic description. In particular, the classical decomposition of the tangent space to the shape space in affine and non affine components [4] is enriched to include also the change in size, in such a way to give a complete description of the tangent space to the size-and-shape space. The proposed generalization is formulated by means of a new Riemannian metric describing the change in size as change in volume rather than change in CS. This lead to a redefinition of some aspect of the Kendalls size-and-shape space without abandoning Kendalls original formulation. This new formulation is discussed by analyzing 3D heart ventricular shapes coming from 3D Speckle Tracking Echocardiography. We evaluate the performances of different methods in recognizing Control (healthy) subjects from patients affected by Hypertrophic Cardiomyopathy.